One of the problems with New Jersey is that so many parts of it are unlike the rest of the state. For example, if you go to the northern portion of the state, you find a lot of industry and cities, and if you go toward the southern half, you find more farms and rural areas. So you’ll get a lot of different answers to your question, since there are so many different areas within the two halves (beach, woods, mountains, cities, farms, and so on).
The climate is temperate. The summers can get very hot and humid, and the winters can get very cold. In NJ, you’ll get it all - freezing rain, snow, thunderstorms, hurricanes, sleet, hail, you name it. There have even been some reports of tornados, although they’re certainly not as widespread as in other areas of the country.
The summer temps can get into the 100s, but rarely. In July and August, you’ll see a lot of 90+ days, and since the humidity is so high, you’ll want to stay indoors, probably. Not sure where you’re coming from, so I don’t know if this is gonna be radically different from what you’re used to.
One thing to look out for that I’ve noticed this year during my last visit (I lived there for 30 years). Many years ago, NJ was invaded by gypsy moth caterpillars. These little bastards are tiny - a couple centimeters - but they’ll eat all of your trees’ leaves. Their last invasion decimated a lot of forestry, mainly because there were so darn many of them. So just a heads up to you for that.
Depending on where you wind up living, public transportation can be feast or famine. If you’re in a city, such as Princeton or Trenton, you will certainly be far better off than if you were in the hinterland, as I was. As with anywhere in the country, the further away from the metropolis you get, the less likely you’ll find good transportation. It’s not as if, contrary to popular opinion, NJ is completely accessible via train or bus from top to bottom, as there are a zillion places that have never seen either form of transportation. During the time I lived there, I used the train maybe three times.
Also, NJ is the most densely populated state, so expect to see a lot of people. And the more people there are, the shorter fuses can be. So don’t get put off if people seem rude; they’re just being themselves! 
Also, car insurance is incredibly high, due in no small part to the amount of people crammed into such a tiny state. If you’re getting new insurance, be aware that it’s mandatory and expensive. Also, companies such as Geico won’t even carry you if you’re a NJ resident.
Whew. Good luck! Wish I had more info for you.